Free product simple question

So I understand abstractly, I think. If we have two groups G, H, then the free product G*H would be the group where elements are finite reduced words of arbitrary length, i.e., powers of elements of g and h, where elements of the same group don't sit next to each other (ex. g^1g^2h^3 is NOT a reduced word because it would be g^3h^3.)

The thing I don't understand then is, if I have say the same group, what would be G*G? Because every combination would just be elements of G which are next to each other. I mean, take for example, g^2g^3. This would reduce to g^5. So aren't I just going to get G again?

Like when I think of the free product of Z * Z. How is this not just Z? Cus a word would be just like 2*5*6*... etc. (finite length). Then every single word would reduce.

I am fairly certain that when we take the free product of a group with itself, we formally think of the elements as being distinct if they come from a different copy of the group. For example, the free product of Z with itself is the free group F_2. It might be useful when doing something like this to mark one copy of the group with a 'dash' e.g write Z*Z' and distinguish the elements of Z' in a similar way.

So what does this look like exactly then? Since the words can be any arbitrary finite length, I can't think of it as having some sort of dimension like I normally would when thinking of say Z X Z. What is this group giving me (Z * Z that is). Just some random combination of integers?

Z is just an infinite cyclic group. So give your copies of Z presentations as follows: Z=<a|-> and Z=<b|-> (the generators are distinct because formally we think of the two copies of Z as being distinct). Then Z*Z=<a,b|-> by the explanation here: http://en.wikipedia.org/wiki/Free_product under "Presentations".
So Z*Z is just the free group on two generators. There's no point in considering what the individual elements of each Z are (or else we will have an image of a bunch of integers randomly thrown together as you say), just that they are infinite cyclic and that we can treat their two generators as being distinct.

oh ok, so it seems like maybe I was thinking of things backwards. This mechanism of using free groups and free products, presentation, etc., is just a way to describe a group. So maybe its not that Z*Z is just something that I want to use for a calculation, maybe its more like, I have some group G, and if I realize some information about it, I might realize its isomorphic to Z*Z, this makes it easier to deal with, etc. This whole process is just kind of a way of describing groups? Am I thinking in the right direction?